March 21, 2022

Competition ….

\[{dN_1 \over dt} = r_1 N \left(1-{N_1 \over K_1} - \alpha {N_2 \over K_1}\right)\]

\[{dN_2 \over dt} = r_2 N \left(1-{N_2 \over K_2} - \beta {N_1 \over K_2}\right)\]

is the outcome of 2 density dependent phenomena:

  1. Intra-specific competition (\(K_1\) and \(K_2\))
  2. Inter-specific competition (\(\alpha\) and \(\beta\))
Patterns depend on the scale and particulars of the competition intensity.

Focus on the the isoclines of zero growth

\[N_2 = {K_1 \over \alpha} - {1\over \alpha} N_1\]

\[N_2 = K_2 - \beta N_1\]

Extinction vs. Equilibria

If:

  • \({1\over\beta} < {K_1 \over K_2} > \alpha\); \(N_1\) wins
  • \({1\over\beta} > {K_1 \over K_2} < \alpha\); \(N_2\) wins
  • \({1\over\beta} < {K_1 \over K_2} < \alpha\); \(N_1\) or \(N_2\) wins
  • \({1\over\beta} > {K_1 \over K_2} > \alpha\); \(N_1\) and \(N_2\) coexist

Equilibria (stable or not) at:

  • \(N^*_1 = {K_1 - \alpha K_2 \over 1-\alpha\beta}\)
  • \(N^*_2 = {K_2 - \beta K_1 \over 1-\alpha\beta}\)

Co-existence essentially occurs when ….

Intra-specific > Inter-specific competition

\[{1\over\beta} > {K_1 \over K_2} > \alpha\]

\(\beta\) and \(\alpha\) small, in a very particular way relative to the ratio of carrying capacities.

Predation

Predation

an ecological process where one organism (the predator) consumes another (the prey).

  • Provides most of the principle route of energy flow through ecosystems
  • Strong selective pressure
  • Chief source of density dependent effects in regulation of many animal (and plant) populations

Predation is any transfer of energy up a trophic level!

  • Herbivory: animals eat plants
    • granivores: eat grain
    • frugivores: eat fruit
  • Parasitism: animals eat other animals without (immediately) killing them
  • Carnivory: animals eat other animals, killing them
  • Cannibalism: animals eat their conspecifics

Basic principles (to model)

  • Growth in predator population depends on the number of prey
  • Growth in prey population depends on the number of predators
  • As an extension of competition
  • An extreme form of competition

Intra-guild predation

Start with competition …

\[{dP \over dt} = r_p P \left(1-{P \over K_p} - \alpha {V \over K_p}\right)\]

\[{dV \over dt} = r_v V \left(1-{V \over K_v} - \beta {P \over K_v}\right)\]

And ADD some extra terms

Species 1 (competitor AND predator \(P\)) benefits:

\[{dP \over dt} = r_p P \left(1-{P \over K_p} - \alpha {V \over K_p}\right) + \gamma VP\]

Species 2 (competitor AND prey \(V\)) suffers:

\[{dV \over dt} = r_v V \left(1-{V \over K_v} - \beta { P\over K_v}\right) - \sigma V P\]

  • \(\gamma\) (gamma) = conversion factor - how much does predators population benefit from eating prey?
  • \(\sigma\) (sigma) = capture efficiency - how much does prey population suffer from getting eaten?

Inspect the isoclines!

\[P^* = K_p - \left(\alpha - {\gamma K_p \over r_p}\right) V\]

\[V^* = K_v - \left(\beta + {\sigma K_v \over r_v}\right) P\]

Qualitatively

  • \(P\) isocline slope becomes STEEPER (and y - intercept increases).
  • \(V\) isocline also becomes STEEPER (and x-intercept decreases)

Rotating isocline

The Lotka-Volterra Predator-Prey Model

That was complicated! Let’s simplify RADICALLY and eliminate ALL competition and density dependence (\(\alpha = \beta = 0\), \(K_v = K_p = \infty\)).

\[{dP \over dt} = -q P + \gamma VP\] \[{dV \over dt} = r V - \sigma VP\]

Predators are always dying at rate \(q\), BUT grow in propotion to prey.

Prey grow exponentially at rate \(r\), BUT die off in proportion to predator.

Assumptions

Lots! And mainly unrealistic!

  • Standard continuous modeling stuff:
    • No age structure
    • Closed population
    • No time lag
  • Instantaneous B & D responses of P to V and V to P
  • Mass action: perfect population mixing in proportion to number of individuals

BUT it’s still a fun model to play with - and a good starting point.

Prey dynamics

\[{dV \over dt} = r V - \sigma VP\]

\(V\) - prey; \(P\) - predators; \(r\) - growth rate; \(\sigma\) - “capture efficiency”

No density dependence (\(K\)), No competition (\(\alpha N_2\)), only removal by \(P\) at rate \(\sigma\).

  • Large \(\sigma\) - strong effect: each moose killed by each wolf has big impact
  • Small \(\sigma\) - weak effect: bats and mosquitoes … hard to make an impact.

Predator dynamics

\[{dP \over dt} = - qP + \gamma VP \]

\(P\) - predators; \(V\) - prey; \(q\) - mortality rate; \(\gamma\) - “conversion” efficiency, e.g. how many prey = new predators

  • High \(\gamma\): high dependency on each discrete prey item, e.g. anaconda and capybara
  • Low \(\gamma\): low dependency on each discrete prey item, e.g. blue whale and krill

Obtain equilibria

Prey equilibrium occurs when

\[P^* = {r\over \sigma}\]

  • Number of predators associated with 0 growth in prey population
  • ratio of prey growth to predator capture efficiency
  • ratio of inputs to outputs

Predator equilibrium occurs when

\[V^* = {q\over \gamma}\]

  • Number of prey associated with 0 growth in predator population
  • ratio of predator death rate to efficiency with which prey becomes predator growth
  • ratio of outputs to inputs

Isoclines!

Prey Isoclines

Predator Isoclines

Graphical analysis

Check out the simulator ….

How does this look?

  • Persistent oscillations with no convergence
  • (1/4 cycle out of phase)
    • \(P_{max}\) and \(P_{min}\) occur when \(V\) is at median
    • \(V_{max}\) and \(V_{min}\) occur when \(P\) is at median

2 exceptions:

  • equilibrium point
  • extreme starting point

Very famous Snowshoe Hare and Lynx dataset

Based (mainly) on fur sales from the Hudson Bay Company in Canada over 100 years. Roughly a 9 to 11 year, fairly synchronous, cycle.

But are these really predator prey oscillations!?

Are predator prey oscillations real?

Are predator prey oscillations real?

… add any carrying capacity to the model:

And we lose all our oscillations!

Are predator prey oscillations real?

Back to the very controlled laboratory environment …

Didinium nausutum vs. Paramecium caudatum.

Georgiy Frantsevich Gause

First attempt

Just put them together and see what happens

Maybe one oscillation?

Second attempt

Add a little prey refuge which Didinium can’t reach.

Paramecia hide until all predators die off, then rebound.

Third attempt

Dribble in 1 Paramecium and 1 Didinium into the dish every 3 days.

Conclusion: Oscillations are not intrinsic to predator prey dynamics, but require some local back-and-forth, immigration - emigration, local dynamics.

Are Hare - Lynx Lotka-Volterra?

Some quibbles

  • Lynx peaks follow Hare peaks!
  • Hare also oscillate where there are no Lynx
  • Peaks are synchronized across North America

Maybe it’s food-web structure?

Hare cycle with vegetation (bottom up control) and multi-species predation (top-down control), whereas Lynx depend strongly just on hare - which solely drive the cycle.

Maybe it’s climate?

Maybe large scale climate oscillations? (sunspots, NAO)

Did we forgot the trappers?

Why do Lynx lag behind Hare? Maybe the Hare eat the Lynx? (HEL hypothesis).

Or … if you add trappers themselves to the equation … you can reshift the cycle.

On balance ….

  • Lotka-Volterra is a nice, but highly theoretical framework … just too many unrealistic assumptions.
  • Like any null model, thinking specifically about WHY it breaks down (which is always) can lead to real insights.
  • Predator-prey phase-spaces is a very useful graphical way to understand dynamic systems.
  • More versatile approaches to predation modeling coming next week…